Abstract
We study the stability of unsteady particle-laden flows in long, tilted water columns in batch settling mode, where the quasi-steady assumption of base flow no longer holds for the fast settling of particles. For this purpose, we introduce a settling time scale in the momentum and transport equations to solve the unsteady base flow, and utilise non-modal analysis to examine the stability of the disturbance flow field. The base flow increases in magnitude as the settling speed decreases and attains its maximum value when the settling speed becomes infinitesimal. The time evolution of the disturbance flow energy experiences an algebraic growth caused by the lift-up mechanism of the wall-normal disturbance, followed by an exponential growth owing to the shear instability of the base flow. The streamwise and spanwise wavenumbers corresponding to the peak energy gain are identified for both stages. In particular, the flow instability is enhanced as the Prandtl number increases, which is attributed to the sharpening of the particle-laden interface. On the other hand, the flow instability is suppressed by the increase in settling speed, because less disturbance energy can be extracted from the base flow. There exists an optimal tilted angle for efficient sedimentation, where the particle-laden flow is relatively stable and is accompanied by a smaller energy gain of the disturbance.
Highlights
Many engineering processes apply tilted liquid columns to accelerate the separation of suspended materials
Concluding remarks In conclusion, we have analysed the stability of particle-laden flows in long, tilted water columns in batch settling mode beyond the quasi-steady approximation, which is essential for the fast settling of particles in the problem of sedimentation
By introducing a settling time scale in the momentum and transport equations, the transient behaviour of base flow was resolved in terms of Reynolds number (Re), Prandtl number (Pr) and the settling
Summary
Many engineering processes apply tilted liquid columns to accelerate the separation of suspended materials. The non-uniform nature in the x-direction of the base flow validates the linear modal stability analysis of spatially evolving disturbances for determining the criteria of wave formation in the quasi-steady state This spatially varied clear-fluid layer can be treated as a case in which particles slowly settle in liquid columns with a continuous settling mode in high-aspect-ratio containers (Herbolzheimer & Acrivos 1981; Davis, Herbolzheimer & Acrivos 1983; Leung & Probstein 1983). Owing to particle settling, the thickness of the clear-fluid layer increases with time, resulting in a change in the upward stream beneath the downward-facing wall, as shown in figure 2(b) These findings regarding the development of interfacial instabilities in high-aspect-ratio containers differ from those of previous studies, those pertaining to cases with fast particle settling (i.e. large particle sizes or low liquid viscosity).
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